Engineering - Materials Selection in Mechanical Design Part 6 pdf

Related case studies Case Study 6.3: Mirrors for large telescopes Case Study 6.4: Table legs There are some very large optical telescopes in the world.. Related case studies Case Study

Materials selection - case studies

6.1 Introduction and synopsis

Here we have a collection of case studies* illustrating the screening methods** of Chapter 5 Each

is laid out in the same way:

(a) the problem statement, setting the scene;

(b) the model, identifying function, objectives and constraints from which emerge the property

(c) the selection in which the full menu of materials is reduced by screening and ranking to a (d) the postscript, allowing a commentary on results and philosophy

Techniques for seeking further information are left to later chapters

The first few examples are simple but illustrate the method well Later examples are less obvious and require clear identification of the objectives, the constraints, and the free variables Confusion here can lead to bizarre and misleading conclusions Always apply common sense: does the selection include the traditional materials used for that application? Are some members of the subset obviously unsuitable? If they are, it is usually because a constraint has been overlooked: it must be formulated and applied

The case studies are deliberately simplified to avoid obscuring the method under layers of detail

In most cases nothing is lost by this: the best choice of material for the simple example is the same

as that for the more complex, for the reasons given in Chapter 5

limits and material indices;

short-list of viable candidates; and

6.2 Materials for oars

Credit for inventing the rowed boat seems to belong to the Egyptians Boats with oars appear in carved relief on monuments built in Egypt between 3300 and 3000 BC Boats, before steam power, could be propelled by poling, by sail and by oar Oars gave more control than the other two, the military potential of which was well understood by the Romans, the Vikings and the Venetians

* A computer-based exploration of these and other case studies can be found in Case Studies in Materials Selection by

M.F Ashby and D Cebon, published by Granta Design, Trumpington Mews, 40B High Street, Trumpington CB2 2LS, UK ( 1996)

**The material properties used here are taken from the CMS compilation published by Granta Design Trumpington Mews, 40B High Street, Trumpington CB2 2LS, UK

Records of rowing races on the Thames in London extend back to 1716 Originally the competitors were watermen, rowing the ferries used to carry people and goods across the river Gradually gentlemen became involved (notably the young gentlemen of Oxford and Cambridge), sophisticating both the rules and the equipment The real stimulus for development of boat and oar came in 1900 with the establishment of rowing as an Olympic sport Since then both have exploited to the full the craftsmanship and materials of their day Consider, as an example, the oar

Mechanically speaking, an oar is a beam, loaded in bending It must be strong enough to carry the bending moment exerted by the oarsman without breaking, it must have just the right stiffness to match the rower’s own characteristics and give the right ‘feel’, and - very important - it must be

as light as possible Meeting the strength constraint is easy Oars are designed on stiffness, that is,

to give a specified elastic deflection under a given load The upper part of Figure 6.1 shows an oar:

a blade or ‘spoon’ is bonded to a shaft or ‘loom’ which carries a sleeve and collar to give positive location in the rowlock The lower part of the figure shows how the oar stiffness is measured: a

10 kg weight is hung on the oar 2.05 m from the collar and the deflection at this point is measured

A soft oar will deflect nearly.50mm; a hard one only 30 A rower, ordering an oar, will specify

how hard it should be

The oar must also be light; extra weight increases the wetted area of the hull and the drag that goes with it So there we have it: an oar is a beam of specified stiffness and minimum weight The

material index we want was derived in Chapter 5 as equation (5.11) It is that for a light, stiff beam:

_ -

Fig 6.1 An oar Oars are designed on stiffness, measured in the way shown in the lower figure, and they must be light

Table 6.1 Design requirements for the oar Function

Objective Minimize the mass Constraints (a) Length L specified

Oar, meaning light, stiff beam

(b) Bending stiffness S specified (c) Toughness G, > 1 kJ/m2

(d) Cost C,,, < $lOO/kg

There are other obvious constraints Oars are dropped, and blades sometimes clash The material must be tough enough to survive this, so brittle materials (those with a toughness less than 1 kJ/m2)

are unacceptable And, while sportsmen will pay a great deal for the ultimate in equipment, there

are limits on cost Given these requirements, summarized in Table 6.1, what materials should make good oars?

The selection

Figure 6.2 shows the appropriate chart: that in which Young’s modulus, E , is plotted against density,

p The selection line for the index M has a slope of 2, as explained in Section 5.3; it is positioned so that a small group of materials is left above it They are the materials with the largest values of M ,

and it is these which are the best choice, provided they satisfy the other constraints (simple property limits on toughness and cost) They contain three classes of material: woods, carbon and glass-fibre reinforced polymers, and certain ceramics (Table 6.2) Ceramics are brittle; their toughnesses fail

to meet that required by the design The recommendation is clear Make your oars out of wood or, better, out of CFRP

to the compression side of the shaft to add stiffness and the blade is glued to the shaft The rough oar is then shelved for some weeks to settle down, and finished by hand cutting and polishing The final spruce oar weigh? between 4 and 4.3 kg, and costs (in 1998) about E150 or $250

Composite blades are a little lighter than wood for the same stiffness The component parts are fabricated from a mixture of carbon and glass fibres in an epoxy matrix, assembled and glued The advantage of composites lies partly in the saving of weight (typical weight: 3.9 kg) and partly in the greater control of performance: the shaft is moulded to give the stiffness specified by the purchaser Until recently a CFRP oar cost more than a wooden one, but the price of carbon fibres has fallen sufficiently that the two cost about the same

Could we do better? The chart shows that wood and CFRP offer the lightest oars, at least when normal construction methods are used Novel composites, not at present shown on the chart, might permit further weight saving; and functional-grading (a thin, very stiff outer shell with a low density core) might do it But both appear, at present, unlikely

Fig 6.2 Materials for oars CFRP is better than wood because the structure can be controlled

Table 6.2 Materials for oars

(GPa)’/’/(Mg/m’)

Woods 5-8 Cheap, traditional, but with natural variability CFRP 4-8 As good as wood, more control of properties GFRP 2-3.5 Cheaper than CFRP but lower M , thus heavier

Ceramics 4-8 Good M but toughness low and cost high

Further reading

Redgrave, S (1992) Complete Book of Rowing, Partridge Press, London

Related case studies

Case Study 6.3: Mirrors for large telescopes

Case Study 6.4: Table legs

There are some very large optical telescopes in the world The newer ones employ complex and cunning tricks to maintain their precision as they track across the sky - more on that in the Postscript But if you want a simple telescope, you make the reflector as a single rigid mirror The largest such telescope is sited on Mount Semivodrike, near Zelenchukskaya in the Caucasus Mountains of Russia The mirror is 6 m (236 inches) in diameter To be sufficiently rigid, the mirror, which is made of glass, is about 1 m thick and weighs 70 tonnes

The total cost of a large (236-inch) telescope is, like the telescope itself, astronomical - about

UK E150m or US $240m The mirror itself accounts for only about 5% of this cost; the rest is that

of the mechanism which holds, positions and moves it as it tracks across the sky This mechanism must be stiff enough to position the mirror relative to the collecting system with a precision about equal to that of the wavelength of light It might seem, at first sight, that doubling the mass m of the mirror would require that the sections of the support structure be doubled too, so as to keep the stresses (and hence the strains and displacements) the same; but the heavier structure then deflects under its own weight In practice, the sections have to increase as m2, and so does the cost Before the turn of the century, mirrors were made of speculum metal (density: about 8 Mg/m3) Since then, they have been made of glass (density: 2.3 Mg/m'), silvered on the front surface, so none

of the optical properties of the glass are used Glass is chosen for its mechanical properties only; the 70tonnes of glass is just a very elaborate support for l00nm (about 30g) of silver Could one,

by taking a radically new look at materials for mirrors, suggest possible routes to the construction

of lighter, cheaper telescopes?

The model

At its simplest, the mirror is a circular disc, of diameter 2a and mean thickness t , simply supported

at its periphery (Figure 6.3) When horizontal, it will deflect under it own weight in; when vertical

it will not deflect significantly This distortion (which changes the focal length and introduces aberrations into the mirror) must be small enough that it does not interfere with performance;

in practice, this means that the deflection 8 of the midpoint of the mirror must be less than the wavelength of light Additional requirements are: high dimensional stability (no creep), and low thermal expansion (Table 6.3)

The mass of the mirror (the property we wish to minimize) is

Fig 6.3 The mirror of a large optical telescope is modelled as a disc, simply supported at its periphery

It must not sag by more than a wavelength of light at its centre

Table 6.3 Design requirements for the telescope mirror Function Precision mirror

Objective Minimize the mass

Constraints (a) Radius n specified

(b) Must not distort more than S under its own weight (c) High dimensional stability: no creep, no moisture take-up, low thermal expansion

312

The lightest mirror is the one with the greatest value of the material index

We treat the remaining constraints as property limits, requiring a melting point greater than 1000 K

to avoid creep, zero moisture take up, and a low thermal expansion coefficient (a -= 20 x 10-6/K)

The selection

Here we have another example of elastic design for minimum weight The appropriate chart is again that relating Young’s modulus E and density p - but the line we now construct on it has a slope of 3, corresponding to the condition M = E ‘ / ’ / p = constant (Figure 6.4) Glass lies on the line M = 2 (GPa)1/3m3/Mg Materials which lie above it are better, those below, worse Glass is much better than steel or speculum metal (that is why most mirrors are made of glass); but it is less

Fig 6.4 Materials for telescope mirrors Glass is better than most metals, among which magnesium is

a good choice Carbon-fibre reinforced polymers give, potentially, the lowest weight of all, but may lack adequate dimensional stability Foamed glass is a possible candidate

Table 6.4 Mirror backing for 200-inch telescope

(GPaj’/’m’/Mg u = 6 m

One must, of course, examine other aspects of this choice The mass of the mirror can be calculated from equation (6.5) for the materials listed in the table Note that the polystyrene foam and the CFRP mirrors are roughly one-fifth the weight of the glass one, and that the support structure could thus be as much as 25 times less expensive than that for an orthodox glass mirror But could they be made?

Some of the choices - the polystyrene foam or the CFRP - may at first seem impractical But the potential cost saving (the factor of 25) is so vast that they are worth examining There are ways

of casting a thin film of silicone rubber or of epoxy onto the surface of the mirror-backing (the polystyrene or the CFRP) to give an optically smooth surface which could be silvered The most obvious obstacle is the lack of stability of polymers - they change dimensions with age, humidity, temperature and so on But glass itself can be reinforced with carbon fibres; and it can also be foamed to give a material with a density not much greater than polystyrene foam Both foamed and carbon-reinforced glass have the same chemical and environmental stability as solid glass They could provide a route to large cheap mirrors

Postscript

There are, of course, other things you can do The stringent design criterion (6 > 1 0 ~ m ) can be partially overcome by engineering design without reference to the material used The 8.2 m Japanese telescope on Mauna Kea, Hawaii and the Very Large Telescope (VLT) at Cerro Paranal Silla in Chile each have a thin glass reflector supported by little hydraulic or piezo-electric jacks that exert distributed forces over its back surface, controlled to vary with the attitude of the mirror The Keck telescope, also on Mauna Kea, is segmented, each segment independently positioned to give optical focus But the limitations of this sort of mechanical system still require that the mirror meet a stiffness target While stiffness at minimum weight is the design requirement, the material-selection criteria remain unchanged

Radio telescopes do not have to be quite as precisely dimensioned as optical ones because they detect radiation with a longer wavelength But they are much bigger (60metres rather than 6) and they suffer from similar distortional problems Microwaves have wavelengths in the mm band, requiring precision over the mirror face of 0.25 mm A recent 45 m radio telescope built for the University of Tokyo achieves this, using CFRP Its parabolic surface is made of 6000 CFRP panels, each servo controlled to compensate for macro-distortion Recent telescopes have been made from CFRP, for exactly the reasons we deduced Beryllium appears on our list, but is impractical for large mirrors because of its cost Small mirrors for space applications must be light for a different reason (to reduce take-off weight) and must, in addition, be as immune as possible to temperature change Here beryllium comes into its own

Related case studies

Case Study 6.5: Materials for table legs

Case Study 6.20: Materials to minimize thermal distortion

6.4 Materials for table legs

Luigi Tavolino, furniture designer, conceives of a lightweight table of daring simplicity: a flat sheet

of toughened glass supported on slender, unbraced, cylindrical legs (Figure 6.5) The legs must be solid (to make them thin) and as light as possible (to make the table easier to move) They must support the table top and whatever is placed upon it without buckling What materials could one recommend?

Fig 6.5 A lightweight table with slender cylindrical legs Lightness and slenderness are independent

design goals, both constrained by the requirement that the legs must not buckle when the table is

loaded The best choice is a material with high values of both E 1 J 2 / p and E

Table 6.5 Design requirements for table legs

Function Column (supporting compressive loads) Objective (a) Minimize the mass

Constraints (a) Length L specified

(b) Maximize slenderness (b) Must not buckle under design loads (c) Must not fracture if accidentally struck

subject to the constraint that it supports a load P without buckling

of length l and radius r (see Appendix A, 'Useful Solutions') is

r 2 E I r3 Er4

e 4t2

Pent = 2 - ~

(6.6) The elastic load Pcfit of a column

using I = r r 4 / 4 where I is the second moment of area of the column The load P must not exceed

P,,,, Solving for the free variable, r , and substituting it into the equation for m gives

The material properties are grouped together in the last pair of brackets The weight is minimized

by selecting the subset of materials with the greatest value of the material index

(a result we could have taken directly from Appendix B)

which will not buckle:

Now slenderness Inverting equation (6.7) with P = P,,, gives an equation for the thinnest leg

The selection

We seek the subset of materials which have high values of E ' / 2 / p and E Figure 6.6 shows the appropriate chart: Young's modulus, E , plotted against density, p A guideline of slope 2 is drawn on the diagram; it defines the slope of the grid of lines for values of E ' / 2 / p The guideline is displaced upwards (retaining the slope) until a reasonably small subset of materials is isolated above it; it

is shown at the position M I = 6GPa'/*/(Mg/m') Materials above this line have higher values of

Fig 6.6 Materials for light, slender legs Wood is a good choice; so is a composite such as CFRP, which, having a higher modulus than wood, gives a column which is both light and slender Ceramics meet the stated design goals, but are brittle

Table 6.6 Materials for table legs

Comment

Cheap, traditional, reliable

CFRP 4-8 30-200 Outstanding M I and M 2 , but expensive

GFRP 3.5-5.5 20-90 Cheaper than CFRP, but lower M I and M 2

Ceramics 4-8 150- 1000 Outstanding M I and M 2 Eliminated by

brittleness

M1 They are identified on the figure: woods (the traditional material for table legs), composites

(particularly CFRP) and certain special engineering ceramics Polymers are out: they are not stiff enough; metals too: they are too heavy (even magnesium alloys, which are the lightest) The choice

is further narrowed by the requirement that, for slenderness, E must be large A horizontal line on the diagram links materials with equal values of E ; those above are stiffer Figure 6.6 shows that placing this line at M 1 = 100 GPa eliminates woods and GFRP If the legs must be really thin, then the shortlist is reduced to CFRP and ceramics: they give legs which weigh the same as the wooden ones but are much thinner Ceramics, we know, are brittle: they have low values of fracture toughness Table legs are exposed to abuse - they get knocked and kicked; common sense suggests that an additional constraint is needed, that of adequate toughness This can be done using Chart 6

(Figure 4.7); it eliminates ceramics, leaving CFRP The cost of CFRP (Chart 14, Figure 4.15) may cause Snr Tavolino to reconsider his design, but that is another matter: he did not mention cost in his original specification

It is a good idea to lay out the results as a table, showing not only the materials which are best, but those which are second-best - they may, when other considerations are involved become the best choice Table 6.6 shows one way of doing it

Postscript

Tubular legs, the reader will say, must be lighter than solid ones True; but they will also be fatter

So it depends on the relative importance Mr Tavolino attaches to his two objectives - lightness and slenderness - and only he can decide that If he can be persuaded to live with fat legs, tubing can

be considered - and the material choice may be different Materials selection when section-shape

is a variable comes in Chapter 7

Ceramic legs were eliminated because of low toughness If (improbably) the goal was to design

a light, blender-legged table for use at high temperatures, ceramics should be reconsidered The

brittleness problem can be by-passed by protecting the legs from abuse, or by pre-stressing them in

compression

Related case studies

Case Study 6.3: Mirrors for large telescopes

Case Study 8.2: Spars for man-powered planes

Case Study 8.3: Forks for a racing bicycle

6.5 Cost - structural materials for buildings

The most expensive thing that most people buy is the house they live in Roughly half the cost of a house is the cost of the materials of which it is made, and they are used in large quantities (family house: around 200 tonnes; large apartment block: around 20 000 tonnes) The materials are used in three ways (Figure 6.7): structurally to hold the building up; as cladding, to keep the weather out; and as ‘internals’, to insulate against heat, sound, and so forth)

Consider the selection of materials for the structure They must be stiff, strong, and cheap Stiff,

so that the building does not flex too much under wind loads or internal loading Strong, so that there is no risk of it collapsing And cheap, because such a lot of material is used The structural frame of a building is rarely exposed to the environment, and is not, in general, visible So criteria

of corrosion resistance, or appearance , are not important here The design goal is simple: strength and stiffness at minimum cost To be more specific: consider the selection of material for floor beams Table 6.7 summarizes the requirements

The model

The way of deriving material indices for cheap, stiff and strong beams was developed in Chapter 5

The results we want are listed in Table 5.7 The critical components in building are loaded either

Fig 6.7 The materials of a building perform three broad roles The frame gives mechanical support; the cladding excludes the environment; and the internal surfacing controls heat, light and sound

Table 6.7 Design requirements for floor beams

Function Floor beams

Objective Minimize the cost

Constraints (a) Length L specified

(b) Stiffness: must not deflect too much under design loads (c) Strength: must not fail unger design loads

in bending (floor joists, for example) or as columns (the vertical members) The two indices that

we want to maximize are:

where, as always, E is Young’s modulus, af is the failure strength, p is the density and C, material cost

The selection

Cost appears in two of the charts Figure 6.8 shows the first of them: modulus against relative cost per unit volume The shaded band has the appropriate slope; it isolates concrete, stone, brick, softwoods, cast irons and the cheaper steels The second, strength against relative cost, is shown in Figure 6.9 The shaded band - M I this time - gives almost the same selection They are listed, with values, in the table They are exactly the materials of which buildings have been, and are, made

Fig 6.9 The selection of cheap, strong materials for the structural frames of buildings

Table 6.8 Structural materials for buildings

concrete

Postscript

It is sometimes suggested that architects live in the past; that in the late 20th century they should

be building with fibreglass (GFRP), aluminium alloys and stainless steel Occasionally they do, but the last two figures give an idea of the penalty involved: the cost of achieving the same stiffness and strength is between 5 and 10 times greater Civil construction (buildings, bridges, roads and the like) is materials-intensive: the cost of the material dominates the product cost, and the quantity used is enormous Then only the cheapest of materials qualify, and the design must be adapted to use them Concrete, stone and brick have strength only in compression; the form of the building must use them in this way (columns, arches) Wood, steel and reinforced concrete have strength both in tension and compression, and steel, additionally, can be given efficient shapes (I-sections, box sections, tubes); the form of the building made from these has much greater freedom

Further reading

Cowan, H.J and Smith, P.R (1988) The Science and Technology ofBuiZding Materials, Van Nostrand-Reinhold,

New York

Related case studies

Case Study 6.2: Materials for oars

Case Study 6.4: Materials for table legs

Case Study 8.4: Floor joists: wood or steel?

6.6 Materials for flywheels

Flywheels store energy Small ones - the sort found in children’s toys - are made of lead Old steam engines have flywheels; they are made of cast iron More recently flywheels have been proposed for power storage and regenerative braking systems for vehicles; a few have been built, some of high-strength steel, some of composites Lead, cast iron, steel, composites - there is a strange diversity here What is the best choice of material for a flywheel?

An efficient flywheel stores as much energy per unit weight as possible, without failing

Failure (were it to occur) is caused by centrifugal loading: if the centrifugal stress exceeds the

tensile strength (or fatigue strength) the flywheel flies apart One constraint is that this should not occur

The flywheel of a child’s toy is not efficient in this sense Its velocity is limited by the pulling- power of the child, and never remotely approaches the burst velocity In this case, and for the flywheel of an automobile engine - we wish to maximize the energy stored per unit volume at

a constant (specified) angular velociv There is also a constraint on the outer radius, R , of the flywheel so that it will fit into a confined space

The answer therefore depends on the application The strategy for optimizing flywheels for efficient energy-storing systems differs from that for children’s toys The two alternative sets of design requirements are listed in Tables 6.9(a) and (b)

The model

An efficient flywheel of the first type stores as much energy per unit weight as possible, without

failing Think of it as a solid disc of radius R and thickness t , rotating with angular velocity o

(Figure 6.10) The energy U stored in the flywheel is

2

Table 6.9(a) Design requirements for maximum-energy flywheel

Function Flywheel for energy storage Objective

Constraints (a) Must not burst

Maximize kinetic energy per unit mass (b) Adequate toughness to give crack-tolerance

Table 6.9(b) Design requirements for limited-velocity flywheel

Function Flywheel for child’s toy Objective

Constraints Outer radius fixed

Maximize kinetic energy per unit volume

Fig 6.10 A flywheel The maximum kinetic energy it can store is limited by its strength

4

m = nR t p (6.12) The quantity to be maximized is the kinetic energy per unit mass, which is the ratio of the last two

equations:

(6.13)

As the flywheel is spun up, the energy stored in it increases, but so does the centrifugal stress The

maximum principal stress in a spinning disc of uniform thickness is

(6.14) where u is Poisson’s ratio This stress must not exceed the failure stress af (with an appropriate

factor of safety, here omitted) This sets an upper limit to the angular velocity, w , and disc radius,

R (the free variables) Eliminating Rw between the last two equations gives

Poissons’s ratio, u, is roughly 1/3 for solids; we can treat it as a constant The best materials for

high-performance flywheels are those with high values of the material index

(6.16)

It has units of kJ/kg

But what of the other sort of flywheel - that of the child’s toy? Here we seek the material which

stores the most energy per unit volume V at constant velocity The energy per unit volume at a

Both R and w are fixed by the design, so the best material is now that with the greatest value of

I

The selection

Figure 6.11 shows Chart 2: strength against density Values of M correspond to a grid of lines of

slope 1 One such line is shown at the value M = 100 H k g Candidate materials with high values

Fig 6.11 Materials for flywheels Composites and beryllium are the best choices Lead and cast iron,

traditional for flywheels, are good when performance is limited by rotational velocity, not strength

of M lie in the search region towards the top left They are listed in the upper part of Table 6.10 The best choices are unexpected ones: beryllium and composites, particularly glass-fibre reinforced polymers Recent designs use a filament-wound glass-fibre reinforced rotor, able to store around

150 kJ/kg; a 20 kg rotor then stores 3 MJ or 800 kWh A lead flywheel, by contrast, can store only

3 kJ/kg before disintegration; a cast-iron flywheel, about 10 All these are small compared with the energy density in gasoline: roughly 20 000 kJ/kg

Even so, the energy density in the flywheel is considerable; its sudden release in a failure could

be catastrophic The disc must be surrounded by a burst-shield and precise quality control in manufacture is essential to avoid out-of-balance forces This has been achieved in a number of

Table 6.10 Materials for flywheels

Ceramics

Composites: CFRP

GFRP Beryllium

200 - 500 100-400

300 100-200

is velocity-limited, not strength-limited

glass-fibre energy-storage flywheels intended for use in trucks and buses, and as an energy reservoir for smoothing wind-power generation

But what of the lead flywheels of children's toys? There could hardly be two more different materials than GFRP and lead: the one, strong and light, the other, soft and heavy Why lead? It is because, in the child's toy, the constraint is different Even a super-child cannot spin the flywheel of his toy up to its burst velocity The angular velocity w is limited, instead, by the drive mechanism (pull-string, friction drive) Then, as we have seen, the best material is that with the largest density (Table 6.10, bottom section) Lead is good Cast iron is less good, but cheaper Gold, platinum and uranium are better, but may be thought unsuitable for other reasons

Postscript

And now a digression: the electric car By the turn of the century electric cars will be on the roads, powered by a souped-up version of the lead-acid battery But batteries have their problems: the energy density they can contain is low (see Table 6.1 1); their weight limits both the range and the performance of the car It is practical to build flywheels with an energy density of roughly five times that of the battery Serious consideration is now being given to a flywheel for electric cars

A pair of counter-rotating CFRP discs are housed in a steel burst-shield Magnets embedded in the discs pass near coils in the housing, inducing a current and allowing power to be drawn to the electric motor which drives the wheels Such a flywheel could, it is estimated, give an electric car

a range of 600 km, at a cost competitive with the gasoline engine

Further reading

Christensen, R.M (1979) Meclzanics of Composite Materials, Wiley Interscience, New York, p 213 et seq

Lewis, G (1990) Selection oj'Enngineering Materials, Prentice Hall, Englewood Cliffs, NJ, Part 1, p I Medlicott, P A C and Potter, K.D ( I 986) The development of a composite flywheel for vehicle applications,

in High Tech - the Way into the Nineties, edited by Brunsch, K., Golden, H-D., and Horkert, C-M Elsevier, Amsterdam, p 29

Table 6.1 1 Energy density of power sources

W/kg

Comment

Gasoline 20 000 Oxidation of hydrocarbon - mass of oxygen

not included

Flywheels Up to 350 Attractive, but not yet proven

Lead-acid battery 40-50 Large weight for acceptable range

Springs rubber bands

agent forms part of fuel

u p to 5 Much less efficient method of energy storage

than flywheel

Related case studies

Case Study 6.7:

Case Study 6.15: Safe pressure vessels

Materials for high-flow fans

6.7 Materials for high-flow fans

Automobile engines have a fan which cools the radiator when the forward motion of the car is insufficient to do the job Commonly, the fan is driven by a belt from the main drive-shaft of the engine The blades of the fan are subjected both to centrifugal forces and to bending moments caused

by sudden acceleration of the motor At least one fatality has been caused by the disintegration of

a fan when an engine which had been reluctant to start suddenly sprang to life and was violently raced while a helper leaned over it What criteria should one adopt in selecting materials to avoid this? The material chosen for the fan must be cheap Any automaker who has survived to the present day has cut costs relentlessly on every component But safety comes first

The radius, R, of the fan is determined by design considerations: flow rate of air, and the

space into which it must fit The fan must not fail The design requirements, then, are those of Table 6.12

The model

A blade (Figure 6.12) has mean section area A and length wR, where w is the fraction of the fan

radius R which is blade (the rest is hub) Its volume is wRA and the angular acceleration is 0 2 R , so

Table 6.12 Design requirements for the fan Function Cooling fan

Objective Constraints (a) Radius R specified

Maximum angular velocity without failure (b) Must be cheap and easy to form

Fig 6.12 A fan The flow-rate of gas through the fan is related to its rotation speed, which is ultimately

limited by its strength

the centrifugal force at the blade root is

F = p(aRA)w2R (6.18) The force is carried by the section A , so the stress at the root of the blade is

(6.19) This stress must not exceed the failure stress C f divided by a safety factor (typically about 3 ) which does not affect the analysis and can be ignored Thus for safety:

Fig 6.13 Materials for cheap high-flow fans Polymers - nylons and polypropylenes - are good; so are die-cast aluminium and magnesium alloys Composites are better, but more difficult to fabricate

in where the fan is placed

Table 6.13 Candidate materials for a high-flow fan

Cheap and easy to cast but poor a j / p

Can be die-cast to final shape

Mouldable and cheap

Lay-up methods too expensive and slow Press from chopped-fibre moulding material

Related case studies

Case Study 6.6:

Case Study 12.2: Forming a fan

Case Study 14.3: A non-ferrous alloy: AI-Si die casting alloys

Materials for flywheels

6.8 Golf-ball print heads

Mass is important when inertial forces are large, as they are in high-speed machinery The golf- ball typewriter is an example: fast positioning of the golf-ball requires large accelerations and decelerations Years before they came on the market, both the golf-ball and the daisy-wheel design had been considered and rejected: in those days print heads could only be made of heavy type-metal, and had too much inertia The design became practical when it was realized that a polymer (density,

1 Mg/m') could be moulded to carry the type, replacing the lead-based type-metal (density, about

10 Mg/m') The same idea has contributed to other high-speed processes, which include printing, textile manufacture, and packaging

The model

A golf-ball print head is a thin-walled shell with the type faces moulded on its outer surface (Figure 6.14) Its outer radius, R , is fixed by the requirement that it carry the usual 88 standard characters; the other requirements are summarized in Table 6.14 The time to reposition it varies as the square root of its mass, m, where

and t is the wall thickness and p the density of the material of which it is made We wish to minimize

this mass The wall thickness must be sufficient to bear the strike force: a force F , distributed over

Fig 6.14 A golf-ball print head It must be strong yet light, to minimize inertial forces during rapid repositioning

Table 6.14 Design requirements for golf-ball print heads

Function Rapidly positioned print head

(d) Can be moulded or cast to give sharply defined type-faces

an area of roughly b2, where b is the average linear dimension of a character When golf-ball print

heads fail, they do so by cracking through the shell wall We therefore require as a constraint that the through-thickness shear stress, F / 4 b t , be less than the failure strength, which, for shear, we approximate by a f / 2 :

The selection

Materials for golf-balls require high a f / p ; then Chart 2 is the appropriate one It is reproduced

in Figure 6.15, with appropriate selection lines constructed on it It isolates two viable classes of candidate materials: metals, in the form of aluminium or magnesium casting alloys (which can

be pressure die-cast) and the stronger polymers (which can be moulded to shape) Both classes, potentially, can meet the design requirements at a weight which is 15 to 20 times less than lead- based alloys which are traditional for type We reject ceramics which are strong in compression but not in bending, and composites which cannot be moulded to give fine detail

Data for the candidates are listed in Table 6.15, allowing a more detailed comparison The final choice is an economic one: achieving high character-definition requires high-pressure moulding techniques which cost less, per unit, for polymers than for metals High-modulus, high-strength polymers become the primary choice for the design

Postscript

Printers are big business: long before computers were invented, IBM was already a large company made prosperous by selling typewriters The scale of the market has led to sophisticated designs Golf-balls and daisy-wheels are made of polymers, for the reasons given above; but not just one

polymer A modern daisy-wheel uses at least two: one for the type-face, which must resist wear

Fig 6.15 Materials for golf-ball print heads Polymers, because of their low density, are better than type-metal, which is mostly lead, and therefore has high inertia

and impact, and a second for the fingers, which act as the return springs Golf-balls have a surface coating for wear resistance, or simply to make the polymer look like a metal Their days, however, are numbered Laser and bubble-jet technologies have already largely displaced them These, too, present problems in material selection, but of a different kind

Related case studies

Case Study 6.6: Materials for flywheels

Case Study 6.7: Materials for high-flow fans

Table 6.15 Materials for golf-ball and daisy-wheel print heads

Type metal

(Pb-5% Sn-10% Sb)

4 15 to 20 times heavier than the above for

the same strength

6.9 Materials for springs

Springs come in many shapes (Figure 6.16) and have many purposes: one thinks of axial springs (a rubber band, for example), leaf springs, helical springs, spiral springs, torsion bars Regardless

of their shape or use, the best material for a spring of minimum volume is that with the greatest value of o ; / E , and for minimum weight it is that with the greatest value of o ; / E p (derived below)

We use them as a way of introducing two of the most useful of the charts: Young’s modulus E

plotted against strength of (Chart 4), and specific modulus, E / p , plotted against specific strength

o f / p (Chart 5 )

The primary function of a spring is that of storing elastic energy and - when required - releasing

it again (Table 6.16) The elastic energy stored per unit volume in a block of material stressed

Fig 6.16 Springs store energy The best material for any spring, regardless of its shape or the way in

which it is loaded, is that with the highest value of or, if weight is important, uF/Ep

Table 6.16 Design requirements for springs Function Elastic spring

Objectives

Constraints

(a) Maximum stored elastic energy per unit volume (b) Maximum stored elastic energy per unit mass (a) No failure by yield, fracture or fatigue (whichever is the most restrictive), meaning CT < c j everywhere in the spring

where E is Young's modulus It is this W , that we wish to maximize The spring will be damaged if

the stress a exceeds the yield stress or failure stress a f ; the constraint is g 5 of So the maximum energy density is

(6.25) Torsion bars and leaf springs are less efficient than axial springs because much of the material is not fully loaded: the material at the neutral axis, for instance, is not loaded at all For torsion bars

The choice of materials for springs of minimum volume is shown in Figure 6.17 A family lines

of slope 1/2 link materials with equal values of M I = ,;/E; those with the highest values of M I

Fig 6.17 Materials for small springs High strength (‘spring’) steel is good Glass, CFRP and GFRP all,

under the right circumstances, make good springs Elastomers are excellent Ceramics are eliminated

by their low tensile strength

lie towards the bottom right The heavy line is one of the family; it is positioned so that a subset

of materials is left exposed The best choices are a high-strength steel ((spring steel, in fact) lying

near the top end of the line, and, at the other end, rubber But certain other materials are suggested too: GFRP (now used for truck springs), titanium alloys (good but expensive), glass (used in galvanometers) and nylon (children’s toys often have nylon springs) Note how the procedure has

identified a candidate from almost every class of material: metals, glasses, polymers, elastomers and composites They are listed, with commentary, in Table 6.17

Table 6.17 Materials for efficient small springs

Brittle in tension; good only in compression

The traditional choice: easily formed and heat treated

Expensive, corrosion-resistant

Comparable in performance with steel; expensive

Almost as good as CFRP and much cheaper

Brittle in torsion, but excellent if protected against damage; very low loss factor

The least good; but cheap and easily shaped, but high loss factor

Better than spring steel; but high loss factor

Materials selection for light

materials with equal values of

springs is shown in Figure 6.18 A family of lines of slope 2 link

One is shown at the value M 2 = 2 M k g Metals, because of their high density, are less good than composites, and much less good than elastomers (You can store roughly eight times more elastic energy, per unit weight, in a rubber band than in the best spring steel.) Candidates are listed in Table 6.18 Wood, the traditional material for archery bows, now appears

Further reading

Boiton, R.G (1963) The mechanics of instrumentation, Proc I Mech E., Vol 177, No 10, 269-288

Hayes, M (1990) Materials update 2: springs, Engineering, May, p 42

Related case studies

Case Study 6.10: Elastic hinges

Case Study 6.12: Diaphragms for pressure actuators

Case Study 8.6: Ultra-efficient springs

Fig 6.18 Materials for light springs Metals are disadvantaged by their high densities Composites are

good; so is wood Elastomers are excellent

Table 6.18 Materials for efficient light springs

Ceramics (5-40) Brittle in tension; good only in compression

Spring steel 2-3 Poor, because of high density

Ti alloys 2-3 Better than steel; corrosion-resistant; expensive

GFRP 3-5 Better than steel; less expensive than CFRP

Glass (fibres) 10-30 Brittle in torsion, but excellent if protected

Wood 1-2 On a weight basis, wood makes good springs

Nylon

Rubber

EP

1.5 -2 20-50

As good as steel, but with a high loss factor

Outstanding; 10 times better than steel, but with high loss factor

6.10 Elastic hinges

Nature makes much use of elastic hinges: skin, muscle, cartilage all allow large, recoverable deflec- tions Man, too, designs with Jlexure and torsion hinges: devices which connect or transmit load between components while allowing limited relative movement between them by deflecting elasti- cally (Figure 6.19 and Table 6.19) Which materials make good hinges?

The model

Consider the hinge for the lid of a box The box, lid and hinge are to be moulded in one operation The hinge is a thin ligament of material which flexes elastically as the box is closed, as in the figure, but it carries no significant axial loads Then the best material is the one which (for given ligament dimensions) bends to the smallest radius without yielding or failing When a ligament of thickness t is bent elastically to a radius R, the surface strain is

t

2R and, since the hinge is elastic, the maximum stress is

Table 6.19 Design requirements for elastic hinges Function

This must not exceed the yield or failure strength a+- Thus the radius to which the ligament can be bent without damage is

(6.30)

The best material is the one that can be bent to the smallest radius, that is, the one with the greatest value of the index

We have assumed thus far that the hinge thickness, t , is dictated by the way the hinge is made

But in normal use, the hinge may also cany repeated axial (tensile) forces, F , due to handling or

to the weight of the box and its contents This sets a minimum value for the thickness, t , which is found by requiring that the tensile stress, F l t w (where w is the hinge width) does not exceed the strength limit af :

The criteria both involve ratios of of and E ; we need Chart 4 (Figure 6.20) Candidates are identified

by using the guide line of slope 1; a line is shown at the position M = a,/E = 3 x lo-* The best choices for the hinge are all polymeric materials The shortlist (Table 6.20) includes polyethylenes, polypropylene, nylon and, best of all, elastomers, though these may be too flexible for the body of the box itself Cheap products with this sort of elastic hinge are generally moulded from polyethylene, polypropylene or nylon Spring steel and other metallic spring materials (like phosphor bronze) are possibilities: they combine usable af / E with high E , giving flexibility with good positional stability (as in the suspensions of relays) The tables gives further details

Postscript

Polymers give more design-freedom than metals The elastic hinge is one example of this, reducing the box, hinge and lid (three components plus the fasteners needed to join them) to a single box- hinge-lid, moulded in one operation Their spring-like properties allow snap-together, easily-joined

Fig 6.20 Materials for elastic hinges Elastomers are best, but may not be rigid enough to meet other

design needs Then polymers such as nylon, PTFE and PE are better Spring steel is less good, but much stronger

parts Another is the elastomeric coupling - a flexible universal joint, allowing an exceptionally high angular, parallel and axial flexibility with good shock absorption characteristics Elastomeric hinges offer many more opportunities, to be exploited in engineering design

Related case studies

Case Study 6.9:

Case Study 6.11 : Materials for seals

Case Study 6.12: Diaphragms for pressure actuators

Materials for springs

Table 6.20 Materials for elastic hinges

1.6-1.8 1.6-1.7 2-2.1 2-2.1 10-20 8-12 10-20

Widely used for cheap hinged bottle caps etc Stiffer than PES Easily moulded

Stiffer than PES Easily moulded

Very durable; more expensive than PE, PP, etc Outstanding, but low modulus

M I less good than polymers Use when high stiffness required

M I less good than polymers Use when high stiffness

required

A reusable elastic seal consists of a cylinder of material compressed between two flat surfaces (Figure 6.21) The seal must form the largest possible contact width, b, while keeping the contact stress, (T sufficiently low that it does not damage the flat surfaces; and the seal itself must remain elastic so that it can be reused many times What materials make good seals? Elastomers - everyone knows that But let us do the job properly; there may be more to be learnt We build the selection around the requirements of Table 6.21

The model

A cylinder of diameter 2R and modulus E , pressed on to a rigid flat surface by a force f per unit

length, forms an elastic contact of width b (Appendix A: 'Useful Solutions') where

(6.31)

This is the quantity to be maximized: the objective function The contact stress, both in the seal and in the surface, is adequately approximated (Appendix A again) by

(6.32) The constraint: the seal must remain elastic, that is, (T must be less than the yield or failure strength,

of, of the material of which it is made Combining the last two equations with this condition gives

The contact width is maximized by maximizing the index

, ,

Fig 6.21 An elastic seal A good seal gives a large conforming contact area without imposing damaging

loads on itself or on the surfaces with which it mates

Table 6.21 Design requirements for the

elastic seals Function Elastic seal Objective Maximum conformability Constraints (a) Limit on contact pressure

(b) low cost

It is also required that the contact stress (T be kept low to avoid damage to the flat surfaces Its value when the maximum contact force is applied (to give the biggest width) is simply af, the failure strength of the seal Suppose the flat surfaces are damaged by a stress of greater than

100 MPa The contact pressure is kept below this by requiring that

M 2 = ~f 5 100MPa

r7

The selection

The two indices are plotted on the mf -E chart in Figure 6.22 isolating elastomers, foams and cork The candidates are listed in Table 6.22 with commentary The value of A 4 2 = 100MPa admits all elastomers as candidates If M 2 were reduced to 10 MPa, all but the most compliant elastomers are eliminated, and foamed polymers become the best bet

Table 6.22 Materials for reusable seals

Butyl rubbers

Polyurethanes 0.5-4.5 Widely used for seals

Silicone rubbers 0.1 -0.8 Higher temperature capability than carbon-chain elastomers,

Nylons 0.05 Near upper limit on contact pressure

Cork 0.1 Low contact stress, chemically stable

Polymer foams

1-3 The natural choice; poor resistance to heat and to some solvents

up to 0.5 Very low contact pressure; delicate seals

the contact pressure; PTFE and silicone rubbers best resist heat and organic solvents The final choice depends on the conditions under which the seal will be used

Related case studies

Case Study 6.9:

Case Study 6.10: Elastic hinges

Case Study 6.12: Diaphragms for pressure actuators

Case Study 6.13: Knife edges and pivots

Materials for springs

A barometer is a pressure actuator Changes in atmospheric pressure, acting on one side of a diaphragm, cause it to deflect; the deflection is transmitted through mechanical linkage or electro- magnetic sensor to a read-out Similar diaphragms form the active component of altimeters, pressure gauges, and gas-flow controls for diving equipment Which materials best meet the requirements for diaphragms, summarized in Table 6.23?

The model

Figure 6.23 shows a diaphragm of radius a and thickness t A pressure difference A p = p1 - p 2

acts across it We wish to maximize the deflection of the centre of the diaphragm, subject to the

Table 6.23 Design requirements for diaphragms

Function Diaphragm for pressure sensing Objective

Constraints

Maximize displacement for given pressure difference (a) Must remain elastic (no yield or fracture) (b) No creep

(c) Low damping for quick, accurate response